Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Let ƒ:[-1,1]→R be continuously differentiable. We consider the question of approximating ƒ′(1) from given data of the form tj,ƒ(tj)^M_(j-1) where the points t_j are in the interval [-1,1]. It is well known that the question is ill-posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order quadrature formulas based on the points t_j's to achieve the approximation. We also estimate the effect of noise on our approximation. The error estimates, both with or without noise, improve upon those in the existing literature, and appear to be unimprovable. The results are applied to the problem of short term prediction of blood glucose concentration, yielding better results than other comparable methods.